Integrand size = 20, antiderivative size = 99 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=-\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a \left (b^2+a c\right ) x^{3/2}+\frac {2}{7} b \left (b^2+6 a c\right ) x^{7/2}+\frac {6}{11} c \left (b^2+a c\right ) x^{11/2}+\frac {2}{5} b c^2 x^{15/2}+\frac {2}{19} c^3 x^{19/2} \] Output:
-2/5*a^3/x^(5/2)-6*a^2*b/x^(1/2)+2*a*(a*c+b^2)*x^(3/2)+2/7*b*(6*a*c+b^2)*x ^(7/2)+6/11*c*(a*c+b^2)*x^(11/2)+2/5*b*c^2*x^(15/2)+2/19*c^3*x^(19/2)
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=-\frac {2 \left (1463 a^3+21945 a^2 b x^2-7315 a b^2 x^4-7315 a^2 c x^4-1045 b^3 x^6-6270 a b c x^6-1995 b^2 c x^8-1995 a c^2 x^8-1463 b c^2 x^{10}-385 c^3 x^{12}\right )}{7315 x^{5/2}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^3/x^(7/2),x]
Output:
(-2*(1463*a^3 + 21945*a^2*b*x^2 - 7315*a*b^2*x^4 - 7315*a^2*c*x^4 - 1045*b ^3*x^6 - 6270*a*b*c*x^6 - 1995*b^2*c*x^8 - 1995*a*c^2*x^8 - 1463*b*c^2*x^1 0 - 385*c^3*x^12))/(7315*x^(5/2))
Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1433, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \int \left (\frac {a^3}{x^{7/2}}+\frac {3 a^2 b}{x^{3/2}}+3 c x^{9/2} \left (a c+b^2\right )+b x^{5/2} \left (6 a c+b^2\right )+3 a \sqrt {x} \left (a c+b^2\right )+3 b c^2 x^{13/2}+c^3 x^{17/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+\frac {6}{11} c x^{11/2} \left (a c+b^2\right )+\frac {2}{7} b x^{7/2} \left (6 a c+b^2\right )+2 a x^{3/2} \left (a c+b^2\right )+\frac {2}{5} b c^2 x^{15/2}+\frac {2}{19} c^3 x^{19/2}\) |
Input:
Int[(a + b*x^2 + c*x^4)^3/x^(7/2),x]
Output:
(-2*a^3)/(5*x^(5/2)) - (6*a^2*b)/Sqrt[x] + 2*a*(b^2 + a*c)*x^(3/2) + (2*b* (b^2 + 6*a*c)*x^(7/2))/7 + (6*c*(b^2 + a*c)*x^(11/2))/11 + (2*b*c^2*x^(15/ 2))/5 + (2*c^3*x^(19/2))/19
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 c^{3} x^{\frac {19}{2}}}{19}+\frac {2 b \,c^{2} x^{\frac {15}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {11}{2}}}{11}+\frac {6 b^{2} c \,x^{\frac {11}{2}}}{11}+\frac {12 a b c \,x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {7}{2}}}{7}+2 a^{2} c \,x^{\frac {3}{2}}+2 a \,b^{2} x^{\frac {3}{2}}-\frac {2 a^{3}}{5 x^{\frac {5}{2}}}-\frac {6 a^{2} b}{\sqrt {x}}\) | \(88\) |
default | \(\frac {2 c^{3} x^{\frac {19}{2}}}{19}+\frac {2 b \,c^{2} x^{\frac {15}{2}}}{5}+\frac {6 a \,c^{2} x^{\frac {11}{2}}}{11}+\frac {6 b^{2} c \,x^{\frac {11}{2}}}{11}+\frac {12 a b c \,x^{\frac {7}{2}}}{7}+\frac {2 b^{3} x^{\frac {7}{2}}}{7}+2 a^{2} c \,x^{\frac {3}{2}}+2 a \,b^{2} x^{\frac {3}{2}}-\frac {2 a^{3}}{5 x^{\frac {5}{2}}}-\frac {6 a^{2} b}{\sqrt {x}}\) | \(88\) |
gosper | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
trager | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
risch | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
orering | \(-\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 b^{2} x^{4} a +21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}}\) | \(90\) |
Input:
int((c*x^4+b*x^2+a)^3/x^(7/2),x,method=_RETURNVERBOSE)
Output:
2/19*c^3*x^(19/2)+2/5*b*c^2*x^(15/2)+6/11*a*c^2*x^(11/2)+6/11*b^2*c*x^(11/ 2)+12/7*a*b*c*x^(7/2)+2/7*b^3*x^(7/2)+2*a^2*c*x^(3/2)+2*a*b^2*x^(3/2)-2/5* a^3/x^(5/2)-6*a^2*b/x^(1/2)
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2 \, {\left (385 \, c^{3} x^{12} + 1463 \, b c^{2} x^{10} + 1995 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 1045 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} - 21945 \, a^{2} b x^{2} + 7315 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 1463 \, a^{3}\right )}}{7315 \, x^{\frac {5}{2}}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="fricas")
Output:
2/7315*(385*c^3*x^12 + 1463*b*c^2*x^10 + 1995*(b^2*c + a*c^2)*x^8 + 1045*( b^3 + 6*a*b*c)*x^6 - 21945*a^2*b*x^2 + 7315*(a*b^2 + a^2*c)*x^4 - 1463*a^3 )/x^(5/2)
Time = 1.21 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=- \frac {2 a^{3}}{5 x^{\frac {5}{2}}} - \frac {6 a^{2} b}{\sqrt {x}} + 2 a^{2} c x^{\frac {3}{2}} + 2 a b^{2} x^{\frac {3}{2}} + \frac {12 a b c x^{\frac {7}{2}}}{7} + \frac {6 a c^{2} x^{\frac {11}{2}}}{11} + \frac {2 b^{3} x^{\frac {7}{2}}}{7} + \frac {6 b^{2} c x^{\frac {11}{2}}}{11} + \frac {2 b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 c^{3} x^{\frac {19}{2}}}{19} \] Input:
integrate((c*x**4+b*x**2+a)**3/x**(7/2),x)
Output:
-2*a**3/(5*x**(5/2)) - 6*a**2*b/sqrt(x) + 2*a**2*c*x**(3/2) + 2*a*b**2*x** (3/2) + 12*a*b*c*x**(7/2)/7 + 6*a*c**2*x**(11/2)/11 + 2*b**3*x**(7/2)/7 + 6*b**2*c*x**(11/2)/11 + 2*b*c**2*x**(15/2)/5 + 2*c**3*x**(19/2)/19
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b c^{2} x^{\frac {15}{2}} + \frac {6}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {7}{2}} + 2 \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="maxima")
Output:
2/19*c^3*x^(19/2) + 2/5*b*c^2*x^(15/2) + 6/11*(b^2*c + a*c^2)*x^(11/2) + 2 /7*(b^3 + 6*a*b*c)*x^(7/2) + 2*(a*b^2 + a^2*c)*x^(3/2) - 2/5*(15*a^2*b*x^2 + a^3)/x^(5/2)
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b c^{2} x^{\frac {15}{2}} + \frac {6}{11} \, b^{2} c x^{\frac {11}{2}} + \frac {6}{11} \, a c^{2} x^{\frac {11}{2}} + \frac {2}{7} \, b^{3} x^{\frac {7}{2}} + \frac {12}{7} \, a b c x^{\frac {7}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} + 2 \, a^{2} c x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \] Input:
integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="giac")
Output:
2/19*c^3*x^(19/2) + 2/5*b*c^2*x^(15/2) + 6/11*b^2*c*x^(11/2) + 6/11*a*c^2* x^(11/2) + 2/7*b^3*x^(7/2) + 12/7*a*b*c*x^(7/2) + 2*a*b^2*x^(3/2) + 2*a^2* c*x^(3/2) - 2/5*(15*a^2*b*x^2 + a^3)/x^(5/2)
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=x^{7/2}\,\left (\frac {2\,b^3}{7}+\frac {12\,a\,c\,b}{7}\right )-\frac {\frac {2\,a^3}{5}+6\,b\,a^2\,x^2}{x^{5/2}}+\frac {2\,c^3\,x^{19/2}}{19}+\frac {2\,b\,c^2\,x^{15/2}}{5}+2\,a\,x^{3/2}\,\left (b^2+a\,c\right )+\frac {6\,c\,x^{11/2}\,\left (b^2+a\,c\right )}{11} \] Input:
int((a + b*x^2 + c*x^4)^3/x^(7/2),x)
Output:
x^(7/2)*((2*b^3)/7 + (12*a*b*c)/7) - ((2*a^3)/5 + 6*a^2*b*x^2)/x^(5/2) + ( 2*c^3*x^(19/2))/19 + (2*b*c^2*x^(15/2))/5 + 2*a*x^(3/2)*(a*c + b^2) + (6*c *x^(11/2)*(a*c + b^2))/11
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {\frac {2}{19} c^{3} x^{12}+\frac {2}{5} b \,c^{2} x^{10}+\frac {6}{11} a \,c^{2} x^{8}+\frac {6}{11} b^{2} c \,x^{8}+\frac {12}{7} a b c \,x^{6}+\frac {2}{7} b^{3} x^{6}+2 a^{2} c \,x^{4}+2 a \,b^{2} x^{4}-6 a^{2} b \,x^{2}-\frac {2}{5} a^{3}}{\sqrt {x}\, x^{2}} \] Input:
int((c*x^4+b*x^2+a)^3/x^(7/2),x)
Output:
(2*( - 1463*a**3 - 21945*a**2*b*x**2 + 7315*a**2*c*x**4 + 7315*a*b**2*x**4 + 6270*a*b*c*x**6 + 1995*a*c**2*x**8 + 1045*b**3*x**6 + 1995*b**2*c*x**8 + 1463*b*c**2*x**10 + 385*c**3*x**12))/(7315*sqrt(x)*x**2)